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Unit 8 · Standard 6.AT.8

Solve and Graph Inequalities

Lesson 8-6 · Learn It — how the math works

Why we're learning this

Detective Kim knows that the suspect withdrew more than $50 from an ATM. The amount plus a $3 fee totaled more than $53. She writes x + 3 > 53 and needs to solve for x, then graph the solution set to share with the bank. How can she find and display all possible withdrawal amounts?

✱ Math Workbench

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How do I solve an inequality and then graph it?

I solve an inequality the same way as an equation: use inverse operations to get the variable alone. But the answer is a range of values, so I show all of them on a number line.

Picture itSee what the math looks like before you read about it

Detective Ortiz is reviewing a stakeout log. The team's rule is that a suspect's vehicle must be parked for more than 20 minutes before they radio for backup. The log lists parked times of 15, 20, and 25 minutes.
Where you see this in real lifeDetective Ortiz is reviewing a stakeout log. The team's rule is that a suspect's vehicle must be parked for more than 20 minutes before they radio for backup. The log lists parked times of 15, 20, and 25 minutes.

Watch me solve itI do — read one step at a time

The problem looks like this
Number line: an open circle at 50 with a ray pointing right shows x > 50; 60 is on the ray, so it checks.404550556065705060
  1. x + 3 > 53. The inverse of adding 3 is subtracting 3.x + 3
  2. I subtract 3 from both sides: x + 3 − 3 > 53 − 3, so x > 50.x + 3
  3. The symbol > does not include 50, so I use an open circle at 50.
  4. 'Greater than' shades right. Check: x = 60 gives 60 + 3 = 63 > 53. True!60 + 3 = 63

Watch out — mistakes students make here

  • '<' is strict, so use an open circle, not a closed one.
  • Subtracting keeps the '<' direction — shade left, not right.
  • Your arithmetic is right, but then compare: is 9 actually greater than 12?

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. x − 5 ≤ 8. What is the inverse of subtracting 5?
  2. We add 5 to both sides: x ≤ 8 + 5, so x ≤ 13.8 + 5
  3. ≤ includes 13, so we use a closed circle and shade left.
  4. x = 10: 10 − 5 = 5 ≤ 8. True?

On your ownYou do — solve this one by yourself

Before you start, remember:

  • and x + 4 > 10, x − 5 ≤ 8, and x + 7 ≥ 15 on the number line.
  • Then finish the practice problems and test a value to check.

Solve: Is x = 5 a to x + 4 > 12?

My answer

Stuck? Open one hint at a time.

Hint 1

'Is 5 a solution?' means: substitute 5 for x and see whether the statement comes out TRUE.

Hint 2

Replace x with 5 and compute the left side completely. Then compare that result to 12 honestly — is it really greater?

Hint 3

Compute 5 + 4 first. Then ask: is that total greater than 12? Your yes-or-no needs the correct REASON attached to it.

Check my answer
✅ Answer: No, because 5 + 4 = 9, and 9 is not greater than 12
Substitute: 5 + 4 = 9. Since 9 > 12 is false, x = 5 is NOT a solution.

Turn & TalkSay it out loud with a partner

Solving an inequality and solving an equation both isolate the variable. What is the same about your steps, and what is different about the answer you get?

Try starting with:

  • The steps are the same because I ___ to ___ the variable.Los pasos son iguales porque ___ para ___ la variable.
  • The answer is different because an inequality has ___.La respuesta es diferente porque una desigualdad tiene ___.
Use these wordssolvegraphsolutionsubstitute